# Classical Indeterminism near Closed Time-like Curves

**Project level:** Honours, PhD

General relativity predicts the possibility of space-time geometries that contain closed time-like curves (CTCs), along which a particle could travel back in time and interact with its past self. Studies of simple bouncing billiard balls along CTCs have revealed a surprising feature: Given the initial position and velocity of the ball, there are typically multiple solutions of the equations of motion, namely many different trajectories compatible with the initial conditions [1]. This result is in striking contrast with the determinism traditionally associated with classical physics and opens the question of what type of predictions are possible in the presence of CTCs.

The aim of this project is to develop a method to make probabilistic predictions for classical systems near CTCs, both considering specific examplesâ€”such as the billiard ball problem [1]â€”and developing a general formalism. The results will be compared with those derived from a quantum-mechanical modelling of the problem [2].

The project requires basic knowledge of the formalism of General Relativity and possibly of the path-integral formulation of quantum mechanics.

[1] F. Echeverria, G. Klinkhammer, and K. S. Thorne, Phys. Rev. D **44**, 1077 (1991).

[2] H. D. Politzer, Phys. Rev. D **49**, 3981 (1994).

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